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\begin{document}

\title{Component Groups of Quotients of $J_0(N)$}
\titlerunning{Component Groups}
\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
\authorrunning{Kohel \and Stein}
\tocauthor{David Kohel (University of Sydney),
William A. Stein (University of California at Berkeley)}
%
\institute{University of Sydney\\
\email{kohel@maths.usyd.edu.au}\\
\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
\and
University of California at Berkeley,\\
\email{was@math.berkeley.edu}\\
\texttt{http://shimura.math.berkeley.edu/\~{}was}
}

\maketitle
\begin{abstract}
Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$ 
be the corresponding optimal abelian variety quotient of $J_0(N)$.  
We describe an algorithm to compute the order of the component group 
of $A_f$ at primes~$p$ that exactly divide~$N$.  We give a table of
orders of component groups for all~$f$ of level $N\leq 127$ and five
examples in which the component group is very large, as predicted 
by the Birch and Swinnerton-Dyer conjecture.
\end{abstract}

\section{Introduction}

Let $X_0(N)$ be the Riemann surface obtained by compactifying the
quotient of the upper half-plane by the action of $\Gamma_0(N)$.
Then $X_0(N)$ has a canonical structure of algebraic curve
over~$\Q$; denote its Jacobian by $J_0(N)$.  It is equipped with
an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of
Hecke operators.  For more details on modular curves, Hecke operators,
and modular forms see, e.g.,~\cite{diamond-im}.

Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform
of weight~$2$ for the congruence subgroup $\Gamma_0(N)$.
The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.
The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$
in a number field $K_f$.  
The kernel $I_f$ of the map $\T \rta R_f$ sending $T_n$ to~$a_n$
is a prime ideal.
Following Shimura~\cite{shimura:factors}, we associate to~$f$ the
quotient $A_f = J_0(N)/I_f J_0(N)$ of
$J_0(N)$.  Then $A_f$ is an abelian variety over~$\Q$ of dimension
$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$.

One-dimensional quotients of $J_0(N)$ have been intensely studied
in recent years, both computationally and theoretically.
The original conjectures of Birch and
Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II},
for elliptic curves over $\Q$, were greatly influenced by
computations.
The scale of these computations was extended and systematized
by Cremona in~\cite{cremona:algs}.

In another direction, Wiles~\cite{wiles:fermat} and
Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
conjecture of Shimura-Taniyama, which asserts that every elliptic
curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them
to establish Fermat's Last Theorem.  The full Shimura-Taniyama
conjecture was later proved by Breuil, Conrad, Diamond, and Taylor
in~\cite{breuil-conrad-diamond-taylor}.
This illustrates the central role played by quotients of $J_0(N)$.

\section{Component Groups of $A_f$}

The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by 
definition a smooth commutative group scheme over~$\Z$ with 
generic fiber~$A$ such that for any smooth scheme~$S$ over~$\Z$, 
the restriction map $$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is 
a bijection.  For more details, including a proof of existence, 
see, e.g.,~\cite{neronmodels}.

Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a
newform~$f$ on $\Gamma_0(N)$, 
and let $\cA_f$ be the N\'eron model of~$A_f$.  
For any prime divisor~$p$ of~$N$, the closed fiber~${\cA_f}_{/\Fp}$ 
is a group scheme over~$\Fp$, which need not be connected.  
Denote the connected component of the identity 
by~${\cA^{\circ}_f}_{/\Fp}$. There is an exact sequence
$$
0 \rightarrow {\cA^{\circ}_f}_{/\Fp}
  \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p}
  \rightarrow 0
$$
with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
called the {\em component group} of $A_f$ at~$p$.

The category of 
finite \'etale group schemes over $\Fp$ is 
equivalent  to  the category of 
finite groups equipped with an action of 
$\Gal(\Fpbar/\Fp)$ (see, e.g., \cite[\S6.4]{waterhouse}).  
The {\em order} of an \'etale group scheme $G/\Fp$ is defined 
to be the order of the group $G(\Fpbar)$.  
In this paper we describe an algorithm for computing the order of 
$\Phi_{A_f,p}$, when~$p$ exactly divides~$N$.

\section{The Algorithm}

Let~$J = J_0(N)$, fix a newform~$f$ of weight-two for $\Gamma_0(N)$,
and let $A_f$ be the corresponding quotient of~$J$.  Because~$J$ is
the Jacobian of a curve, it is canonically isomorphic to its dual, so
the projection $J \rta A_f$ induces a polarization $A_f^{\vee} \rta
A_f$, where $A_f^{\vee}$ denotes the abelian variety dual of $A_f$.
We define the {\em modular degree} $\delta_{A_f}$ of $A_f$ to be the
positive square root of the degree of this polarization.  This agrees
with the usual notion of modular degree when $A_f$ is an elliptic curve.

A {\it torus} $T$ over a field $k$ is a group scheme whose base 
extension to the separable closure $k_s$ of $k$ is a finite 
product of copies of $\G_m$.  Every commutative algebraic group 
over~$k$ admits a unique maximal subtorus, defined over~$k$, 
whose formation commutes with base extension (see IX \S2.1 
of~\cite{groth:sga7}).  The {\it character group} 
of a torus~$T$ is the group $\cX = \Hom_{k_s}(T,\G_m)$ which is 
a free abelian group of finite rank together with an action of 
$\Gal(k_s/k)$ (see, e.g.,~\cite[\S7.3]{waterhouse}).

We apply this construction to our setting as follows. 
The closed fiber of the N\'eron model of $J$ at~$p$ is a group 
scheme over $\Fp$, whose maximal torus we denote by $T_{J,p}$. 
We define $\cX_{J,p}$ to be the character group of $T_{J,p}$. 
Then $\cX_{J,p}$ is a free abelian group equipped with an 
action of both $\Gal(\Fpbar/\Fp)$ and the Hecke algebra~$\T$ 
(see, e.g.,~\cite{ribet:modreps}).  
Moreover, there exists a bilinear pairing
$$
\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z
$$
called the {\em monodromy pairing} such that
$$
\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)).
$$
Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$
for~$t$
in $I_f$, and let
$$
\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z)
$$
be the map induced by the monodromy pairing.
The following theorem of the second author~\cite{stein:phd}, provides the
basis for the computation of orders of component groups.

\begin{theorem}\label{thm:main}
With the notation as above, we have the equality
$$
\#\Phi_{A_f,p}
   = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}}
          {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,.
$$
\end{theorem}

\subsection{Computing the modular degree $\delta_{A,f}$}
Using modular symbols (see, e.g.,~\cite{cremona:algs}), 
we first compute the homology group $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
Using lattice reduction, we compute the $\Z$-submodule 
$H_1(X_0(N),\Z;\mbox{\rm cusps})$
generated by all Manin symbols $(c,d)$.  Then 
$H_1(X_0(N),\Z)$ is the
{\em integer} kernel of the boundary map.  

The Hecke ring $\T$ acts on $H_1(X_0(N),\Z)$ and also on 
the linear dual $\Hom(H_1(X_0(N),\Z),\Z)$,
where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$ 
by $(t.\varphi)(x) = \varphi(tx)$.
We have a natural restriction map
$$
r_f:\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow
   \Hom(H_1(X_0(N),\Z)[I_f],\Z).
$$
\begin{proposition}
The cokernel of $r_f$ is isomorphic to the kernel
of the polarization $A_f^{\vee}\rightarrow A_f$
induced by the map $J_0(N)\rightarrow A_f$.
\end{proposition}

Thus the order of the cokernel of $r_f$ is the square of the modular
degree~$\delta_f$.  We pause to note that the degree of any
polarization is a square; see, e.g.,~\cite[Thm.~13.3]{milne:abvar}.
\begin{proof}
Let $S = S_2(\Gamma_0(N),\C)$ be the complex vector space of 
weight-two modular forms of level~$N$, and set $H = H_1(X_0(N),Z)$. 
The integration pairing $S\times H \rightarrow \C$
induces a natural map
$$\Phi_f:H\rightarrow \Hom(S[I_f],\C).$$
Using the classical
Abel-Jacobi theorem, we deduce the following commutative diagram,
which has exact columns, but whose rows are not exact.
$$\[email protected]=.5cm{
    0\ar[d]            &        0\ar[d]             &  0\ar[d]   \\
  H[I_f]\ar[d]\ar[r] &  H\ar[d]\ar[r]&\Phi_f(H)\ar[d] \\
\Hom(S,\C)[I_f]\ar[d]\ar[r] &\Hom(S,\C)\ar[d]\ar[r] &\Hom(S[I_f],\C)\ar[d]\\
  {A_f^{\vee}(\C)}\ar[d]\ar[r]
   \[email protected]/_3.5pc/[rr]& J_0(N)(\C)\ar[d]\ar[r]& A_f(\C)\ar[d]\\
    0   &   0    &  0   \\
}$$
By the snake lemma, 
the kernel of $A_f^{\vee}(\C)\rightarrow A_f(\C)$ is isomorphic to the
cokernel of the map $H[I_f] \rightarrow \Phi_f(H)$.
Since
    $$\Hom(H/\ker(\Phi_f),\Z) \isom \Hom(H,\Z)[I_f],$$
the $\Hom(-,\Z)$ dual of the map $H[I_f] \rightarrow \Phi_f(H)=H/\ker(\Phi_f)$
is $r_f$, which proves the proposition.
\end{proof}

\subsection{Computing the character group $\cX_{J,p}$}
Let $N = Mp$, where $M$ and $p$ are coprime.  If~$M$ is small, 
then the algorithm of Mestre and Oesterl\'e~\cite{mestre:graphs} 
can be used to compute $\cX_{J,p}$.  This algorithm constructs 
the graph of isogenies between $\Fpbar$-isomorphism classes of 
pairs consisting of a supersingular elliptic curve and a cyclic 
$M$-torsion subgroup.  In particular, the method is elementary 
to apply when $X_0(M)$ has genus~$0$.

In general, the above category of ``enhanced'' supersingular
elliptic curves can be replaced by one of left (or right) ideals 
of a quaternion order~$\cO$ of level~$M$ in the quaternion 
algebra over~$\Q$ ramified at~$p$.
This gives an equivalent category, in which the computation of
homomorphisms is efficient.  The character group $\cX_{J,p}$ is
known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
isomorphic to the degree zero subgroup $\cX(\cO)$ of the free
abelian ``divisor group'' on the isomorphism classes of enhanced
supersingular  elliptic curves and of quaternion ideals.  
Moreover, this isomorphism  is compatible with the operation of 
Hecke operators, which are  effectively computable in $\cX(\cO)$ 
in terms of ideal homomorphisms.

The inner product of two classes in this setting is defined
to be the number of isomorphisms between any two representatives.
The linear extension to $\cX(\cO)$ gives an inner product which
agrees, under the isomorphism, with the monodromy pairing on
$\cX_{J,p}$.  This gives, in particular, an isomorphism $\Phi_{J,p}
\isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an
effective means of computing $\#\coker(\alpha_f)$ and
$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.

The arithmetic of quaternions has been implemented in
{\sc Magma}~\cite{magma} by the first author.  Additional details
and the application to Shimura curves, generalizing $X_0(N)$,
can be found in Kohel~\cite{kohel}.

\subsection{The Galois action on $\Phi_{A_f,p}$}

To determine the Galois action on $\Phi_{A_f,p}$, we need only
know the action of the Frobenius automorphism $\Frob_p$.  
However, $\Frob_p$ acts on $\Phi_{A_f,p}$ in the same way
as $-W_p$, where $W_p$ is the 
$p$th Atkin-Lehner involution, which can be computed using modular 
symbols.  Since~$f$ is an eigenform, the involution $W_p$ acts 
as either $+1$ or $-1$ on $\Phi_{A_f,p}$.
Moreover, the operator $W_p$ is determined by an involution on
the set of quaternion ideals, so it can be determined explicitly
on the character group.

\section{Tables}

The main computational results of this work are presented below
in two tables.  The relevant algorithms have been implemented in
{\sc Magma} and will be made part of a future release.  
They can also be obtained from the second author.

\subsection{Component groups at low level}

Table~\ref{tbl:lowlevel} gives the component groups of the 
quotients $A_f$ of $J_0(N)$ for $N\leq 127$.
The column labeled $d$ contains the
dimensions of the $A_f$, 
and the column labeled $\#\Phi_{A_f,p}$ contains a list
of the orders of the component groups of $A_f$, 
one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
An entry of ``?'' indicates that $p^2\mid N$, so our algorithm
does not apply.
A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is 
nontrivial.  More data along these lines can be obtained from 
the second author.

\begin{table}
\begin{center}
\caption{Component groups at low level}
\end{center}
\label{tbl:lowlevel}
$$
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
11 & 1 & 5\\
14 & 1 & 6^*,3\\
15 & 1 & 4^*,4\\
17 & 1 & 4\\
19 & 1 & 3\\
20 & 1 & ?,2^*\\
21 & 1 & 4,2^*\\
23 & 2 & 11\\
24 & 1 & ?,2^*\\
26 & 1 & 3^*,3\\
 & 1 & 7,1^*\\
27 & 1 & ?\\
29 & 2 & 7\\
30 & 1 & 4^*,3,1^*\\
31 & 2 & 5\\
32 & 1 & ?\\
33 & 1 & 6^*,2\\
34 & 1 & 6,1^*\\
35 & 1 & 3^*,3\\
   & 2 & 8,4^*\\
36 & 1 & ?,?\\
37 & 1 & 1^*\\
   & 1 & 3\\
38 & 1 & 9^*,3\\
   & 1 & 5,1^*\\
39 & 1 & 2^*,2\\
   & 2 & 14,2^*\\
40 & 1 & ?,2\\
41 & 3 & 10\\
42 & 1 & 8,2^*,1^*\\
43 & 1 & 1^*\\
   & 2 & 7\\
44 & 1 & ?,1^*\\
45 & 1 & ?,1^*\\
46 & 1 & 10^*,1\\
47 & 4 & 23\\
48 & 1 & ?,2\\
49 & 1 & ?\\
50 & 1 & 1^*,?\\
   & 1 & 5,?\\
51 & 1 & 3,1^*\\
   & 2 & 16^*,4\\
52 & 1 & ?,2^*\\
53 & 1 & 1^*\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
 & 3 & 13\\
54 & 1 & 3^*,?\\
 & 1 & 3,?\\
55 & 1 & 2,2^*\\
 & 2 & 14^*,2\\
56 & 1 & ?,1\\
 & 1 & ?,1^*\\
57 & 1 & 2^*,1^*\\
 & 1 & 2,2^*\\
 & 1 & 10,1^*\\
58 & 1 & 2^*,1^*\\
 & 1 & 10,1^*\\
59 & 5 & 29\\
61 & 1 & 1^*\\
 & 3 & 5\\
62 & 1 & 4,1^*\\
 & 2 & 66^*,3\\
63 & 1 & ?,1^*\\
 & 2 & ?,3\\
64 & 1 & ?\\
65 & 1 & 1^*,1^*\\
 & 2 & 3^*,3\\
 & 2 & 7,1^*\\
66 & 1 & 2^*,3,1^*\\
 & 1 & 4,1^*,1^*\\
 & 1 & 10,5,1\\
67 & 1 & 1\\
 & 2 & 1^*\\
 & 2 & 11\\
68 & 2 & ?,2^*\\
69 & 1 & 2,1^*\\
 & 2 & 22^*,2\\
70 & 1 & 4,2^*,1^*\\
71 & 3 & 5\\
 & 3 & 7\\
72 & 1 & ?,?\\
73 & 1 & 2\\
 & 2 & 1^*\\
 & 2 & 3\\
74 & 2 & 9^*,3\\
 & 2 & 95,1^*\\
75 & 1 & 1^*,?\\
 & 1 & 1,?\\
 & 1 & 5,?\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
76 & 1 & ?,1^*\\
77 & 1 & 2^*,1^*\\
 & 1 & 3^*,2\\
 & 1 & 6,3^*\\
 & 2 & 2,2^*\\
78 & 1 & 16^*,5^*,1\\
79 & 1 & 1^*\\
 & 5 & 13\\
80 & 1 & ?,2\\
 & 1 & ?,2^*\\
81 & 2 & ?\\
82 & 1 & 2^*,1^*\\
 & 2 & 28,1^*\\
83 & 1 & 1^*\\
 & 6 & 41\\
84 & 1 & ?,1^*,2^*\\
 & 1 & ?,3,2\\
85 & 1 & 2^*,1\\
 & 2 & 2^*,1^*\\
 & 2 & 6,1^*\\
86 & 2 & 21^*,3\\
 & 2 & 55,1^*\\
87 & 2 & 5,1^*\\
 & 3 & 92^*,4\\
88 & 1 & ?,1^*\\
 & 2 & ?,2^*\\
89 & 1 & 1^*\\
 & 1 & 2\\
 & 5 & 11\\
90 & 1 & 2^*,?,3\\
 & 1 & 6,?,1^*\\
 & 1 & 4,?,1\\
91 & 1 & 1^*,1^*\\
 & 1 & 1,1\\
 & 2 & 7,1^*\\
 & 3 & 4^*,8\\
92 & 1 & ?,1^*\\
 & 1 & ?,1\\
93 & 2 & 4^*,1^*\\
 & 3 & 64,2^*\\
94 & 1 & 2,1^*\\
 & 2 & 94^*,1\\
95 & 3 & 10,2^*\\
 & 4 & 54^*,6\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
96 & 1 & ?,2\\
 & 1 & ?,2^*\\
97 & 3 & 1^*\\
 & 4 & 8\\
98 & 1 & 2^*,?\\
 & 2 & 14,?\\
99 & 1 & ?,1^*\\
 & 1 & ?,1\\
 & 1 & ?,1^*\\
 & 1 & ?,1^*\\
100 & 1 & ?,?\\
101 & 1 & 1^*\\
 & 7 & 25\\
102 & 1 & 2^*,2^*,1^*\\
 & 1 & 6^*,6,1^*\\
 & 1 & 8,4,1\\
103 & 2 & 1^*\\
 & 6 & 17\\
104 & 1 & ?,1^*\\
 & 2 & ?,2\\
105 & 1 & 1,1,1\\
 & 2 & 10^*,2^*,2\\
106 & 1 & 4^*,1^*\\
 & 1 & 5^*,1\\
 & 1 & 24,1^*\\
 & 1 & 3,1^*\\
107 & 2 & 1^*\\
 & 7 & 53\\
108 & 1 & ?,?\\
109 & 1 & 1\\
 & 3 & 1^*\\
 & 4 & 9\\
110 & 1 & 7^*,1^*,3\\
 & 1 & 3,1^*,1^*\\
 & 1 & 5,5,1\\
 & 2 & 16^*,3,1^*\\
111 & 3 & 10^*,2\\
 & 4 & 266,2^*\\
112 & 1 & ?,1^*\\
 & 1 & ?,1\\
 & 1 & ?,1^*\\
113 & 1 & 2\\
 & 2 & 2\\
 & 3 & 1^*\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
 & 3 & 7\\
114 & 1 & 2^*,5^*,1\\
 & 1 & 20,3^*,1^*\\
 & 1 & 6,3,1\\
115 & 1 & 5^*,1\\
 & 2 & 4^*,1^*\\
 & 4 & 32,4^*\\
116 & 1 & ?,1^*\\
 & 1 & ?,2^*\\
 & 1 & ?,1^*\\
117 & 1 & ?,1\\
 & 2 & ?,3\\
 & 2 & ?,1^*\\
118 & 1 & 2^*,1^*\\
 & 1 & 19^*,1\\
 & 1 & 10,1^*\\
 & 1 & 1,1^*\\
119 & 4 & 9,3^*\\
 & 5 & 48^*,8\\
120 & 1 & ?,1,1^*\\
 & 1 & ?,2,1\\
121 & 1 & ?\\
 & 1 & ?\\
 & 1 & ?\\
 & 1 & ?\\
122 & 1 & 4^*,1^*\\
 & 2 & 39^*,3\\
 & 3 & 248,1^*\\
123 & 1 & 1^*,1^*\\
 & 1 & 5,1\\
 & 2 & 7,1^*\\
 & 3 & 184^*,4\\
124 & 1 & ?,1^*\\
 & 1 & ?,1\\
125 & 2 & ?\\
 & 2 & ?\\
 & 4 & ?\\
126 & 1 & 8^*,?,1^*\\
 & 1 & 2,?,1\\
127 & 3 & 1^*\\
 & 7 & 21\\
&&\\
&&\\
&&\\
\end{array}$$
\end{table}


\subsection{Examples of large component groups}

Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
J.~Tate in~\cite{tate:bsd}.   The second author computed
the rational numbers $L(A_f,1)/\Omega_{A_f}$ for every
newform~$f$ of level $N\leq 1500$.
The five largest prime divisors occur in the ratios
given in Table~\ref{table:lratios}.
The Birch and Swinnerton-Dyer conjecture predicts that the large
prime divisor of the numerator of each special value must 
divide the order either of some component group $\Phi_{A_f,p}$ or of the 
Shafarevich-Tate group of~$A_f$.  In each instance
$\Phi_{A_f,2}$ is divisible by the large prime divisor, as
predicted.

\begin{table}
\label{table:lratios}
\begin{center}
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
\end{center}
$$\begin{array}{ccll}
 \qquad N \qquad\quad &
 \quad \dim \quad &
 \quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad &
 \qquad \#\Phi_{A_f,p} \\
       1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2
      & 2^?\tdot 17^2 \tdot 85495047371,  2^? \\
       1238=2\tdot 619  & 19 & 2^?\tdot 7553329019/5\tdot 31
      & 2^?\tdot 5\tdot31\tdot7553329019 ,  2^?\\
       1322=2\tdot 661  & 21 & 2^?\tdot 57851840099/331
      & 2^?\tdot 331 \tdot 57851840099, 2^?\\
       1382=2\tdot 691  & 20 & 2^?\tdot 37 \tdot 1864449649 /173
      & 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\
       1478=2\tdot 739  & 20 &
            2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37
      & 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\
\end{array}$$
\end{table}

\section{Further directions}

Further considerations are needed to compute the {\em group}
structure of $\Phi_{A_f,p}$.  However, since the action of Frobenius
is known, computing the group structure of $\Phi_{A_f,p}$ suffices
to determine its structure as a group scheme.

%An equivalence with quaternion divisor groups is not known to
%hold for the character group at~$p$ when $p^2$ divides~$N$.
%Thus 
Our methods say nothing about the component group at primes
whose {\em square} divides the level.  The free abelian group
on classes of nonmaximal orders of index~$p$ at a ramified prime
gives a well-defined divisor group.
Do the resulting Hecke modules determine the component groups
for quotients of level $p^2M$?

Is it possible to define quantities as in Theorem~\ref{thm:main}
even when the weight of~$f$ is {\em greater than~$2$}?
If so, how are the resulting quantities related to the Bloch-Kato
Tamagawa  numbers (see~\cite{bloch-kato}) of the higher weight 
motive attached to~$f$?

\newpage
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{1}

\bibitem{birch-swd-I}
B.\thinspace{}J.~Birch, and
   H.\thinspace{}P.\thinspace{}F.~Swinnerton-Dyer,
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\end{document}


Oh, and the tables now get pushed into the middle of 
the references.  If you have a quick and dirty way 
of fixing this, then please do so as I find it ugly. 
Hmmm, I played slightly with this to no success, but 
also noted that I get no table numbers, at least as 
it compiles on the system here.  Could you look at 
this?  Sorry for leaving you this dirty work. 

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Typographical errors or syntax:

- page 1:  "-" between One and dimensional?
- Sec 2 line 1/2: mistaken \par inserted?
- Sec 3.1 before display: mistaken \par inserted?
- sentence on pagebreak 3/4: do you mean canonically isomorphic
  instead of canonically equivalent?  That would help to interpret the
  next sentence.
- page 4 isomophism (maybe spell check the whole document?)
- Sec 3.3 last sentence: "so can" --> "so it can"
- Sec 4.2 "either the order of" --> "the order of either" ?

Corrected.

Comments of substance: 

- last line of sec 2:
  how can two objects in distinct categories be equivalent?
  Maybe say that GIVING one is equivalent to GIVING the other.

We correct this by saying that the (implicit) parent categories 
are equivalent rather than two objects in them. 

- Section 3: can you explain or give a reference for what the
  "toric part" is.  The average ANTS reader would appreciate it.
  It is not a subgroup or a quotient, but a subquotient, right?

We add a short paragraph to define a torus and the existence 
of a maximal torus, with references.

- Sec 3.1 last sentence: You view abelian varieties as
  complex lattices?  That sounds like a stretch--or do you have
  some equivalence in mind.  Maybe view their fundamental groups
  as complex lattices?

We rewrote this section for clarity, and added a proof of the
formula for the modular degree.

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